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Introduction to the mathematics of general relativity
About: Introduction to the mathematics of general relativity is a research topic. Over the lifetime, 2583 publications have been published within this topic receiving 73295 citations.
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TL;DR: In this article, it is shown that the Riemann curvature tensor has twenty independent components, ten of which appear in the Weyl tensor, and nine of these components appearing in the Einstein tensor.
Abstract: In a four-dimensional curved space-time it is well-known that the Riemann curvature tensor has twenty independent components; ten of these components appear in the Weyl tensor, and nine of these components appear in the Einstein curvature tensor. It is also known that there are fourteen combinations of these components which are invariant under local Lorentz transformations. In this paper, we derive explicitly closed form expressions which contain these twenty independent components in a manifest way. We also write the fourteen invariants in two ways; firstly, we write them in terms of the components; and, secondly, we write them in a covariant fashion, and we further derive the appropriate characteristic value equations and the corresponding Cayley-Hamilton equations for these invariants. We also show explicitly how all of the relevant components transform under a Lorentz transformation. We shall follow the very general and powerful methods of Sachs [1]. We shall not point out at every stage of the calculation which equations are due to Sachs, and which equations are new; this is easily ascertained. Generally speaking, however, the equations depending on the Einstein curvature tensor, and the emphasis placed on this tensor, appear to be new.
20 citations
TL;DR: In this article, four boundary conditions allowing an unambiguous definition of angular momentum of a Cauchy data set in general relativity are presented, and four sets of boundary conditions for general relativity.
Abstract: Four sets of boundary conditions allowing an unambiguous definition of angular momentum of a Cauchy data set in general relativity are presented.
20 citations
TL;DR: In this paper, a model of the gravitational field based on two symmetric tensors is presented, and the equations of motion of test particles are derived: massive particles do not follow a geodesic but massless particles trajectories are null geodesics of an effective metric.
20 citations
TL;DR: In this article, a first-order purely frame-formulation of general relativity is obtained in the gauge natural bundle framework, where a new space is introduced and a first order purely frameformulation is obtained.
Abstract: In the gauge natural bundle framework, a new space is introduced and a first-order purely frame-formulation of general relativity is obtained.
20 citations
TL;DR: A solution to the Einstein field equations that represents a rigidly rotating dust accompanied by a thin matter shell of the same type is found in this article, where the same authors also present a solution to a similar problem in the case of a thin shell.
Abstract: A solution to the Einstein field equations that represents a rigidly rotating dust accompanied by a thin matter shell of the same type is found.
19 citations